Structure vs. Randomness for Bilinear Maps

TL;DR

This paper establishes the equivalence of various notions of rank for 3-tensors, leading to new insights into the complexity of bilinear maps and resolving open questions in the field.

Contribution

It proves the equivalence of slice, analytic, and geometric ranks for 3-tensors, providing new bounds and settling open problems.

Findings

Ranks are equivalent up to a constant for 3-tensors.

Strong trade-offs on arithmetic complexity of biased bilinear maps.

Resolved open questions on tensor ranks and bilinear map complexity.

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.