Structure vs. Randomness for Bilinear Maps

Abstract: We prove that the slice rank of a 3-tensor (a combinatorial notion introduced by Tao in the context of the cap-set problem), the analytic rank (a Fourier-theoretic notion introduced by Gowers and Wolf), and the geometric rank (an algebro-geometric notion introduced by Kopparty, Moshkovitz, and Zuiddam) are all equal up to an absolute constant. As a corollary, we obtain strong trade-offs on the arithmetic complexity of a biased bilinear map, and on the separation between computing a bilinear map exactly and on average. Our result settles open questions of Haramaty and Shpilka [STOC 2010], and of Lovett [Discrete Anal. 2019] for 3-tensors.