Subrank and Optimal Reduction of Scalar Multiplications to Generic Tensors

Since the seminal works of Strassen and Valiant it has been a central theme in algebraic complexity theory to understand the relative complexity of algebraic problems, that is, to understand which algebraic problems (be it bilinear maps like matrix multiplication in Strassen's work, or the determinant and permanent polynomials in Valiant's) can be reduced to each other (under the appropriate notion of reduction). In this paper we determine precisely how many independent scalar multiplications can be reduced to a given bilinear map (this number is called the subrank, and extends the concept of matrix diagonalization to tensors), for essentially all (i.e. generic) bilinear maps. Namely, we prove for a generic bilinear map $T : V \times V \to V$ where $\dim(V) = n$ that $\theta(\sqrt{n})$ independent scalar multiplications can be reduced to $T$. Our result significantly improves on the previous upper bound from the work of Strassen (1991) and B\"urgisser (1990) which was $n^{2/3 + o(1)}$. Our full result is much more general and applies not only to bilinear maps and 3-tensors but also to $k$-tensors, for which we find that the generic subrank is $\theta(n^{1/(k-1)})$. Moreover, as an application we prove that the subrank is not additive under the direct sum. The subrank plays a central role in several areas of complexity theory (matrix multiplication algorithms, barrier results) and combinatorics (e.g., the cap set problem and sunflower problem). 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).